共振条件下一类方程无界解和周期解的共存性

黄国安,刘期怀

(1.桂林航天工业高等专科学校计算机系,广西桂林 541004;2.广西师范大学数学科学院,广西桂林 541004;3.苏州大学数学科学学院,江苏苏州 215006)

共振条件下一类方程无界解和周期解的共存性

黄国安1,2,刘期怀3

(1.桂林航天工业高等专科学校计算机系,广西桂林 541004;2.广西师范大学数学科学院,广西桂林 541004;3.苏州大学数学科学学院,江苏苏州 215006)

讨论了在共振条件下一类具有等时位势的方程无界解和周期解的共存性.利用Poincar´e映射轨道的性质,给出了无界解的存在性条件.在此条件下,由Poincar´e-Bohl定理,得到了方程的一个周期解,进而说明共振条件下这类方程无界解和周期解的是可以共存的.最后,给出了一个无界解和周期解共存的具有等时位势的方程实例.

等时位势；无界解；周期解；共振

1 引言

共振条件下的无界解和周期解的共存性引起了许多数学研究者的兴趣,并已获得了丰富的结果[1-3].本文考虑一类具有等时位势的方程

这里f(x)∈C(R,R),V(x)为2π/n−等时位势(n∈N),ϕ(t,x)∈C(R×R,R)关于t是2π-周期且有界的.称V(x)为T-等时位势,即x00+V0(x)=0的所有解为T-周期的.

当V0(x)=ax+−bx−,ϕ(t,x)=−g(x)+p(t)时,在非共振条件下,利用反转系统的Moser扭转定理,文[4]给出了方程所有解有界f(x),g(x)应满足的一些条件.对于一般的ϕ(t,x),随后作者又给出了非共振条件下该方程有界解和周期解共存的条件[5].因此,一个自然的问题是,在共振条件下,对于方程(1)的解是否满足相应的有界解和周期解共存性？本文参照文[1-2,6]中的讨论方法并对该问题给出了答案.

2 一类平面映射无界轨道的存在性

为了证明我们的主要结果,先给出一类平面映射轨道的性质.

给定σ＞0,令

3 主要结果及其证明

对任意的ρ＞0,记φ(·,ρ)为方程x00+V0(x)=0满足φ(0,ρ)=ρ,φt(0,ρ)=0的解,ψ(t)为方程x00+ax+−bx−=0满足x(0)=1,x0(0)=0的解,这里x+=max{x,0},x−=max{−x,0}, 则ψ(t)为2π/n-周期的且

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Coexistence of unbounded and periodic solutions to a class of isochronous oscillators at resonance

HUANG Guo-an1,2,LIU Qi-huai3
(1.Department of Computer Guilin College of Aerospace Technology,Guilin541004,China; 2.Department of Mathematics Guangxi Normal University,Guilin541004,China; 3.Department of Mathematics Suzhou University,Suzhou215006,China)

This paper studies the coexistence of unbounded and periodic solutions to a class of isochronous oscillators at resonance,obtaining the conditions of existence of unbounded solutions by the dynamics of Ponica´e mapping.Under these conditions,we can get a periodic solution using Ponica´e-Bohl theorem.Thus,it implies the coexistence of unbounded and periodic solutions.Finially,an example for the equations which has the coexistence of unbounded and periodic solutions is given.

isochronous potential,unbounded solutions,periodic solutions,resonance

O175.2

A

1008-5513(2009)03-0603-07

2008-09-26.

黄国安(1971-),讲师,在读硕士,研究方向：常微分方程与金融工程.

2000MSC:34C11,34C25